PART 2: Stokes's Theorem & Magnetic Flux and Magnetic Flux Density
I. Stokes’s Theorem
⃗ = 𝑟 cos ∅𝑎𝑟 + sin ∅𝑎∅ and
1. Verify both sides of Stokes’s theorem for the vector field 𝐵
the path shown in the following Fig. 1.
Fig. 1.
Solution:
⃗ ∙ 𝑑𝑙
(a) First, let evaluate the LHS: ∮𝐿 𝐵
1
⃗ ) ∙ 𝑑𝑆
(b) Now, let evaluate the RHS: ∫𝑆(∇ × 𝐵
➢ Hence, Stokes’s Theorem is verified.
⃗ = cos ∅𝑎𝑟 + sin ∅𝑎∅ and
2. Verify both sides of Stokes’s theorem for the vector field 𝐵
the path shown in the following Fig. 2.
Fig. 2
Solution:
⃗ ∙ 𝑑𝑙
(a) First, let evaluate the LHS: ∮𝐿 𝐵
2
⃗ ) ∙ 𝑑𝑆
(b) Now, let evaluate the RHS: ∫𝑆(∇ × 𝐵
➢ Hence, Stokes’s Theorem is verified.
⃗ = 𝑟 cos ∅𝑎𝑟 + sin ∅𝑎∅ and
3. Verify both sides of Stokes’s theorem for the vector field 𝐵
the path shown in the following Fig. 2. [Solve it!]
Fig. 2
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4. Evaluate both sides of Stokes’s theorem for the field 𝐺 = 10 sin 𝜃𝑎∅ and the
surface 𝑟 = 3, 0 ≤ 𝜃 ≤ 900 , 0 ≤ ∅ ≤ 900. Let the surface have the 𝑎𝑟 direction.
Solution:
(a) First, let evaluate the LHS: ∮𝐿 𝐺 ∙ 𝑑𝑙
(Note: You must elaborate all the steps involved)
(b) Now, let evaluate the RHS: ∫𝑆(∇ × 𝐺 ) ∙ 𝑑𝑆
(Note: You must elaborate all the steps involved)
➢ Hence, Stokes’s Theorem is verified.
1
∅
∅
⃗ = cos ( ) 𝑎𝜌 − sin ( ) 𝑎∅ A⁄m, evaluate both sides of
5. Given the field 𝐻
2
2
2
Stokes’s theorem for the path formed by the intersection of the cylinder 𝜌 = 3
and the plane 𝑧 = 2, and for the surface defined by 𝜌 = 3, 0 ≤ 𝑧 ≤ 2, and 𝑧 =
0, 0 ≤ 𝜌 ≤ 3.
Solution:
⃗ ∙ 𝑑𝑙
(a) First, let evaluate the LHS: ∮𝐿 𝐻
(Note: You must elaborate all the steps involved)
(b) Now, let evaluate the RHS: ∫𝑆(∇ × 𝐺 ) ∙ 𝑑𝑆
(Note: You must elaborate all the steps involved)
➢ Hence, Stokes’s Theorem is verified.
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II. Magnetic Flux and Magnetic Flux Density
NOTE: You must elaborate all the steps involved in all of the given solutions
⃗ =
1. The magnetic field intensity is given in a certain region of space as 𝐻
2
𝑧
𝑥+2𝑦
𝑧2
𝑎𝑦 +
𝑎𝑧 A⁄m. Find:
⃗
(a) ∇ × 𝐻
(b) Use 𝐽 to find the total current passing through the surface 𝑧 = 4, 1 ≤ 𝑥 ≤ 2, 3 ≤ 𝑧 ≤
5, in the 𝑎𝑧 direction.
Solution:
(a)
(b) The total current….
2. A solid nonmagnetic conductor of circular cross section has a radius of 2 mm. The
conductor is inhomogeneous, with 𝜎 = 106 (1 + 106 𝜌2 ) S⁄m. if the conductor is 1 m in
length and has a voltage of 1 mV between its ends, find:
⃗
(a) 𝐻
(b) The total magnetic flux inside the conductor.
Solution:
(a) The magnetic intensity …
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(b) The total magnetic flux
3. The cylindrical shell defined by 1 cm < 𝜌 < 1.4 cm consists of a nonmagnetic conducting
material and carries a total current of 50 A in the 𝑎𝑧 direction. Find the total magnetic flux
crossing the plane ∅ = 0, 0 < 𝑧 < 1:
(a) 0 < 𝜌 < 1.2 cm
(b) 1.0 cm < 𝜌 < 1.4 cm
(c) 1.4 cm < 𝜌 < 20 cm
Solution:
(a) Magnetic flux …
(b) Magnetic flux…
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(c) Magnetic flux…
4. The free-space region defined by 1 < 𝑧 < 4 cm and 2 < 𝜌 < 3 cm is a toroid of
⃗ =
rectangular cross section. Let the surface at 𝜌 = 3 cm carry a surface current 𝐾
2𝑎𝑧 kA⁄m.
(a) Specify the current on the surfaces at 𝜌 = 2 cm, 𝑧 = 1 cm, and 𝑧 = 4 cm.
⃗ everywhere.
(b) Find 𝐻
(c) Calculate the total flux within the toroid.
Solution:
(a) The current…
(b) Magnetic Field intensity
(c) Total Flux
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